Eigenstate Correlations in Dual-Unitary Quantum Circuits: Partial Spectral Form Factor

TL;DR

This paper computes the partial spectral form factor (PSFF) for chaotic dual-unitary quantum circuits, linking eigenstate correlations to random-matrix theory. Using a tensor-network representation and space-transfer operators, the PSFF is evaluated exactly in the thermodynamic limit: it remains constant at short times due to locality, and at longer times it matches the random-matrix (CUE) prediction up to exponentially small corrections, with a precise bound |K_A(t) − K_A^{CUE}(t)| ≤ c D_A t^2 d^{-t}. The method hinges on a reduction to a finite-dimensional problem via permutation-based Weingarten calculus, enabling exact results and robust numerical confirmation for both finite systems and the thermodynamic limit. The findings strengthen the view of dual-unitary circuits as paradigmatic models of quantum chaos and open avenues for probing higher PSFF moments and related entanglement-focused diagnostics in many-body quantum dynamics.

Abstract

While the notion of quantum chaos is tied to random matrix spectral correlations, also eigenstate properties in chaotic systems are often assumed to be described by random matrix theory. Analytic insights into eigenstate correlations can be obtained by the recently introduced partial spectral form factor. Here, we study the partial spectral form factor in chaotic dual-unitary quantum circuits in the thermodynamic limit. We compute the latter for a finite subsystem in a brickwork circuit coupled to an infinite complement. For initial times, shorter than the subsystem's size, spatial locality and (dual) unitarity implies a constant partial spectral form factor, clearly deviating from the linear ramp of the random matrix prediction. In contrast, for larger times we prove, that the partial spectral form factor follows the random matrix result up to exponentially suppressed corrections. We supplement our exact analytical results by semi-analytic computations performed in the thermodynamic limit as well as with numerics for finite-size systems.

Figures (3)

• Figure 1: Schematic representation of the PSFF $K_A(t)$ for a subsystem $A$ of dimension $D_A$ in random matrix theory, i.e., for the CUE of dimension $D$ (solid black line) and our results in dual-unitary circuits (red line) with $d$-dimensional local Hilbert spaces. The latter include exact analytic results for initial times (solid red line), numerical results obtained in the thermodynamic limit for transient times (dotted red line), exact results for later times (solid red line) valid in the thermodynamic limit $D \to \infty$, as well as numerical data for late times $t>D$ obtained in finite systems (dashed red line). We indicate both the offset (thin gray dashed line) and the linear ramp of the PSFF. Only the leading behavior in $D_A$ and the dimension of the complement, $D_{A^{\mathrm{c}}}$, is shown.
• Figure 2: PSFF $K_A(t)$ for dual-unitary circuits with $d=2$, $2L=12$ and $J=0.2$ for subsystem size $2l=6$ (red crosses) and $2l=2$ (blue circles). The black line corresponds to the random matrix result $K_A^{\mathrm{CUE}}(t;D)$ with only the leading terms in $1/D$ kept. Only times which are a multiple of 64 are shown. Time is measured in units of Heisenberg time $D$ and the vertical axis is shifted and rescaled in order to compare different subsystem sizes. The inset shows the corresponding initial time regime without rescaling time by $D$ and for all times. Results are obtained from averaging over at least $10^4$ realizations.
• Figure 3: Rescaled PSFF for dual-unitary circuits with $d=2$ in the thermodynamic limit for $l=1$ (red crosses) and $l=10$ (blue circles). The random matrix result $K_A^{\mathrm{CUE}}(t)$ is depicted as a black line.